Elements and Applications of Integral Equations (UMAP)

This module deals with the Volterra and Fredholm integral equations and their applications. In addition to problems that can be directly modeled by differential equations and then reduced to integral equations are discussed. The major methods, namely numerical and iterative, for solving integral equations are illustrated.

Table of Contents:

1. INTRODUCTION
1.1 Integral Equations as a Natural Setting
1.2 Volterra and Fredholm Integral Equations
1.3 The Method of Laplace Transform

2. PROBLEMS REDUCED TO INTEGRAL EQUATIONS
2.1 Initial Value Problems Reduced to Volterra Integral Equations
2.2 Boundary Value Problems Reduced to Fredholm Integral Equations
2.3 From Integral to Differential Equations - Leibniz's Rule

3. SOME BASIC METHODS OF SOLUTIONS
3.1 The Method of Successive Approximations
3.2 A Numerical Method

4. MORE PROBLEMS AS INTEGRAL EQUATIONS
4.1 Mortality of Equipment and Its Rate of Replacement
4.2 Torsion of a Wire
4.3 The Automatic Control of a Rotating Shaft
4.4 Small Deflection of a Rotating Shaft
4.5 Optimized Shape of Electrical Signal
4.6 The Bernoulli Problem
4.7 The Laplace and Other Integral Transforms
4.8 Abel's Problem
4.9 The Electrified Disc-Dual Integral Equations

5. MODEL EXAM

6. ANSWERS TO EXERCISES

7. ANSWERS TO MODEL EXAM

8. GENERAL REFERENCE